This is directly MGSS Mazda Service Global Support: Some customers may experience the check engine light ON with DTC P0126:00 (Thermostat stuck open) and one or more of the following symptoms: The engine takes longer to warm up The engine temperature gauge fluctuates The low coolant temperature indicator (blue) light stays on longer or comes on while driving Poor heater performance, especially at idle

Your going to need a OBD code reader diagnostic tool , it’s a device that you hook up to vehicles OBD port. It communicates with the vehicles computer. Go to AutoZone, they will scan the code for free.

They should cover the repair under goodwill. You should stand your ground. Don’t back down. Its a common issue, there is even a TSB

Thermostat could be stuck in the open position. P0126 fault code. There is TSB for this.